Kallåk, VetleParametric Design and Analysis of Wave-Induced Responses of a Semi-Submersible Floating Wind Turbine Platform NTNU Norwegian University of Science and Technology Faculty of Engineering Department of Marine Technology
Master ’s thesis
Parametric Design and Analysis of Wave-Induced Responses of a Semi- Submersible Floating Wind Turbine Platform
Master’s thesis in Marine Technology Supervisor: Bachynski, Erin
June 2020
Parametric Design and Analysis of Wave-Induced Responses of a Semi- Submersible Floating Wind Turbine Platform
Master’s thesis in Marine Technology Supervisor: Bachynski, Erin
June 2020
Norwegian University of Science and Technology Faculty of Engineering
Department of Marine Technology
tions and internal hull loads of a semi-submersible floating 10 MW wind turbine platform. The purpose of the investigation is to optimize the hull form to minimize cost, while complying with strength and safety requirements. The need for more sustainable energy sources is increasing.
Offshore wind brings several advantages over onshore, and recent studies show that the potential for harvesting offshore wind energy is huge. Most of this potential comes from deep water sites, where floating structures are advantageous. However, for floating wind to compete with other energy sources it needs to be cost-competitive. Therefore, a parametric study is carried out to better understand how the design parameters influence the responses of the structure and how the design could be improved.
The WindFloat semi-submersible platform was chosen as the baseline design for the study and upscaled to support a 10 MW reference wind turbine. In this study, the hull was parameterized by defining four main dimensions: 1) the column diameter, 2) the draft, 3) the distance between the columns, and 4) the heave plate size. Also, the effect of a wind-induced mean tilt angle on the responses was investigated.
The first-order potential theory programWadam was used to calculate the hydrodynamic added mass, damping, and excitation loads on the columns and heave plates. The mass and restoring properties were calculated in MATLAB. To simplify calculations, the motions and internal hull loads were calculated in the frequency domain. The hydrodynamic loads on the truss mem- bers connecting the columns and viscous loads on the heave plates were included by linearizing Morison’s equation. Evaluating one parameter at the time, the influence on the hydrostatic properties and wave-induced responses was studied. The standard deviations of the motions, nacelle accelerations, and internal hull loads for each design and three environmental conditions were used to compare the designs.
The most significant finding from the parametric study shows that a 28 % reduction in steel mass can be achieved by: 1) reduce the column diameter and heave plate size, 2) increase the distance between the columns, and 3) reduce the draft. Further, the improved design has lower internal loads, similar motions, and slightly higher nacelle accelerations compared to the initial upscaled design. A smaller column diameter and a larger distance between the columns reduces the steel mass while still counteracting the overturning moment from the wind turbine thrust.
Increasing the distance between the columns did not result in a higher bending moment in the cross-section. It should be noted that a rigid body is assumed, and the truss members were not included in the sectional load analysis. A smaller column diameter increases the heave and pitch motions and nacelle accelerations. However, the heave plates could then also be reduced in size while keeping the heave natural period outside the wave energy range. Smaller heave plates decrease motions and nacelle accelerations more than by increasing the column diameter.
Both reducing the column diameter and heave plate size decrease the cross-sectional bending moment. Increasing the draft is the most efficient parameter for reducing the motions and na- celle acceleration, at the cost of increased steel mass and higher split forces between the columns.
The most important finding from the mean tilt analysis shows a 31 % increase in the nacelle acceleration standard deviation when analyzing the structure with a -10 degrees tilt angle. The potential wave loads are sensitive to the vertical displacement of the heave plates when the platform is tilted, especially at higher frequencies in the wave energy range. The heave force was doubled for a wave frequency of one rad/s when the platform was tilted 10 degrees.
elsene og de interne lastene fra bølger til en halvt nedsenkbar flytende 10 MW vindturbin. Hen- sikten med undersøkelsen er å optimere skroget med tanke på kostnad, mens krav til styrke og sikkerhet er overholdt. Det er et økende behov for fornybar energi. Havvind har flere fordeler i forhold til landbasert, og nyere studier viser at potensialet er enormt. Mesteparten av dette potensialet er fra områder med dypt vann, hvor flytende strukturer er fordelaktig. For at fly- tende havvind skal kunne konkurrere med andre energikilder må det være konkurransedyktig på pris. En parameterstudie er derfor gjennomført for å bedre forstå hvordan design parameterne påvirker oppførselen til strukturen og hvordan designet kan forbedres.
Den halvt nedsenkbare WindFloat plattformen ble valgt som grunndesign og ble oppskalert for å bære en 10 MW referanse vindturbin. I studien har skroget blitt gjort parametrisk ved å definere fire hoveddimensjoner: 1) søylediameteren, 2) dypgangen, 3) avstanden mellom søylene og 4) størrelsen på hivplatene. I tillegg har effekten av en vindindusert krengevinkel på oppførselen til plattformen blitt undersøkt.
Førsteordens, potensialteori programmetWadam ble brukt for å beregne hydrodynamisk tilleg- gsmasse, demping og eksitasjons laster på søylene og hivplatene. Masse- og stivhetsegenskapene ble beregnet iMATLAB. For å forenkle beregningene ble bevegelsene og de interne lastene bereg- net i frekvensdomenet. De hydrodynamiske lastene på bjelkene mellom søylene og de viskøse kreftene på hivplatene ble inkludert ved linearisering av Morisons likning. Ved å evaluere en parameter av gangen, ble påvirkningen på de hydrostatiske og bølgeinduserte responsene til strukturen studert. Standardavviket til bevegelsene, nacelle akselerasjonene og de interne las- tene for hvert design og tre ulike sjøtilstander ble brukt for å sammenligne plattformene.
Den viktigste funnet fra parameterstudien er at stålmassen kan reduseres med 28 % ved å: 1) redusere søylediameteren og hivplate størrelsen, 2) øke avstanden mellom søylene, og 3) redusere dypgangen. Det forbedrede designet har lavere interne laster, like bevegelser og litt høyere aksel- erasjoner i nacellen i forhold til det første, oppskalerte designet. En lavere søylediameter og større avstand mellom søylene reduserer stålvekten, mens plattformen opprettholder nok stivhet til å motvirke momentet fra vindturbinen. Å øke avstanden mellom søylene ga ikke større bøyemo- ment i tverrsnittet. Det skal nevnes at stivt legeme er antatt og bjelkene mellom søylene var ikke med i beregningene av interne laster. En mindre søylediameter øker hiv og stamp bevegelsene og nacelle akselerasjonene. Imidlertid, kan da også hivplatene reduseres i størrelse mens hiv egen- perioden holdes utenfor bølgeenergi området. Mindre hivplater minker bevegelsene og nacelle akselerasjonene mer enn ved å øke søylediameteren. Både mindre søylediameter og hivplater reduserer bøyemomentet i tverrsnittet. Å øke dypgangen er det mest effektive for å redusere bevegelsene og nacelle akselerasjonene, på bekostning av mer stål og høyere splittkrefter mellom søylene.
Det viktigste funnet fra analysene med krengevinkel ga 31 % økning i standardavviket til nacelle akselerasjonene når en krengevinkel på -10 grader ble brukt. Potensial bølgekreftene er følsomme for endringer i dypgangen til hivplatene når plattformen krenger, spesielt ved høyere frekvenser i bølgeenergi området. Kraften i hiv-retning ble doblet ved en bølgefrekvens på en rad/s når plattformen ble krenget 10 grader.
Technology at the Norwegian University of Science and Technology (NTNU). The master’s thesis is written during the spring semester of 2020. I have found the topic of floating wind turbines very interesting and see great potential for this technology. Through the parametric study, I have gained a deeper understanding of the hydrodynamics of floating wind turbine hulls, which I hope will be valuable knowledge in my future work career.
Special gratitude goes out to Professor Erin Bachynski at NTNU for guidance and supervising during the master’s thesis work the spring of 2020. Our weekly meetings, first at the office and then online due to the COVID-pandemic, are much appreciated. I am very grateful for the knowledge Professor Bachynski has shared about floating wind turbines and the use of differ- ent software. I am also thankful for the discussions and feedback I have received about the thesis.
I would also like to thank Jørgen Kvaleid, COWI, for introducing the topic of floating wind turbines and for sharing valuable information concerning general hydrodynamics related to the topic. I hope to be able to explore this topic further in COWI.
Vetle Kallåk
Trondheim, June 9th, 2020
NTNU Trondheim
Norwegian University of Science and Technology
Department of Marine Technology – Group of Marine Structures
MASTER THESIS IN MARINE TECHNOLOGY SPRING 2020
FOR
STUD.TECHN. Vetle Kallåk
Parametric design of semi-submersible floating wind turbines Parametrisk design av halvt nedsenkbare flyttende vindturbiner
Background:
The wind industry has developed very fast in recent years, moving from onshore to offshore in shallow water and then to floating wind turbines (FWTs) in deep water. Semi-submersible floating wind turbines are a promising technology due to their wide applicability across different water depths and relatively simple installation. A wide variety of designs have been proposed, and there is increasing interest in optimization of the hull form to minimize cost, while still satisfying important strength and safety requirements.
In order to better understand the consequences of changes in the hull design on the overall responses of semi-submersible floating wind turbines, a parametric study of the hydrodynamic loads and wave- induced responses is proposed. For one (or more) parametrized hull shapes, several different floater designs for a 10 MW wind turbine will be proposed, and the effects of the hull shape on the first order response amplitude operator (RAO) will be assessed. In addition to global motions, first order wave loads within the hull should also be considered.
Assignment:
The following tasks should be addressed:
1. Literature review regarding semi-submersible floating wind turbine designs, response analysis, and design criteria. Review of theoretical considerations related to first and second order potential flow theory.
2. Identification of one (or more) hull form(s), and a baseline design for a 10 MW wind turbine.
Definition of design parameters and design criteria.
3. Development of parametric inputs to i.e. Genie and HydroD for generating hull geometry and computing first order wave loads. Comparison of results for the baseline design against literature.
4. Parametric studies regarding influence of design parameters on the first order response amplitude operators (RAOs) for motions, and for internal hull loads. The effect of mean tilt angle on the RAOs should be considered.
5. If time permits, carry out global analysis of two selected designs in SIMA (including wind loads), and assess the importance of the wave loads compared to wind loads.
6. Report and conclude on the investigation.
The work scope could be larger than anticipated. Subject to approval from the supervisor, topics may be deleted from the list above or reduced in extent.
In the project, the candidate shall present his personal contribution to the resolution of problem within the scope of the project work.
Theories and conclusions should be based on mathematical derivations and/or logic reasoning identifying the various steps in the deduction.
2
The candidate should utilize the existing possibilities for obtaining relevant literature.
The project report should be organized in a rational manner to give a clear exposition of results, assessments, and conclusions. The text should be brief and to the point, with a clear language.
Telegraphic language should be avoided.
The project report shall contain the following elements: A text defining the scope, preface, list of contents, main body of the project report, conclusions with recommendations for further work, list of symbols and acronyms, reference and (optional) appendices. All figures, tables and equations shall be numerated.
The supervisor may require that the candidate, in an early stage of the work, present a written plan for the completion of the work. The plan should include a budget for the use of computer and laboratory resources that will be charged to the department. Overruns shall be reported to the supervisor.
The original contribution of the candidate and material taken from other sources shall be clearly defined. Work from other sources shall be properly referenced using an acknowledged referencing system.
Erin Bachynski Supervisor
Deadline: June 10th 2020
List of Figures viii
List of Tables x
Nomenclature xi
1 Introduction 1
1.1 Background . . . 1
1.2 Scope and Objectives . . . 1
1.3 Structure of the Report . . . 2
2 Literature Review 3 2.1 Offshore Wind Industry . . . 3
2.1.1 Floating Wind Turbines . . . 3
2.2 Existing Semi-Submersible Designs . . . 4
2.2.1 Olav Olsen OO-Star . . . 5
2.2.2 NAUTILUS . . . 5
2.2.3 WindFloat . . . 6
2.2.4 Fukushima . . . 6
2.2.5 OC4-DeepCwind . . . 7
2.2.6 Comparison of Existing Designs . . . 7
2.3 Related Studies . . . 7
2.4 Upscaling Procedure . . . 9
2.5 Effects of Heave Plates . . . 10
3 Theory 13 3.1 Structural Dynamics . . . 13
3.1.1 Rigid Body Equations of Motions . . . 13
3.1.2 Frequency Domain Method . . . 14
3.2 Hydrodynamics . . . 15
3.2.1 Linear Wave Theory . . . 15
3.2.2 Statistical Description of Waves and Responses . . . 16
3.2.3 Hydrostatic Restoring Forces and Moments . . . 16
3.2.4 First-Order Effects . . . 17
3.2.5 Second-Order Effects . . . 18
3.2.6 Viscous Forces . . . 18
3.2.7 Morison’s Equation . . . 18
3.2.8 Added Mass and Drag Coefficients . . . 20
4 Method 22 4.1 Choice and Definitions of FWT Floater Baseline Design . . . 22
4.2 Procedure . . . 23
4.2.1 GeniE . . . 23
4.2.2 HydroD and Wadam . . . 24
4.2.3 MATLAB . . . 24
4.2.4 Internal Hull Loads . . . 25
4.2.5 Mean Tilt Angle . . . 28
4.3 10 MW Reference Wind Turbine . . . 28
4.4 Environmental Conditions . . . 30
4.5 Design Criteria . . . 31
4.5.4 Other Criteria . . . 32
4.6 Design Parameters . . . 33
4.6.1 Main Dimensions . . . 33
4.6.2 General Assumptions about the Design . . . 34
4.6.3 Assumptions for the Analyses . . . 35
4.7 Verification of Procedure . . . 36
4.7.1 General . . . 36
4.7.2 Comparison of Baseline Design against Literature . . . 36
4.7.3 Verification of Internal Hull Loads Calculations . . . 40
5 Results and Discussion 42 5.1 Upscaled Design . . . 42
5.1.1 Mass and Hydrostatic Properties . . . 42
5.1.2 Hydrodynamic Results . . . 43
5.1.3 Internal Hull Loads for the Upscaled Design . . . 46
5.2 Effects of Parameter Variation on Platform Behavior . . . 48
5.2.1 Column Diameter . . . 48
5.2.2 Draft . . . 55
5.2.3 Distance between Columns . . . 60
5.2.4 Length of Heave Plates . . . 65
5.2.5 Summary . . . 69
5.3 Improved Design . . . 73
5.4 Effects of Mean Tilt Angle . . . 76
6 Conclusion and Recommendations 84 6.1 Conclusion . . . 84
6.2 Recommendations for Further Work . . . 85
7 References 86
A Upscaled Design I
B Parameter Variation II
B.1 Column Diameter . . . II B.2 Draft . . . V B.3 Distance between Columns . . . VII B.4 Length of Heave Plates . . . VIII B.5 Mean Tilt Angle . . . X
C Estimation of Heave Excitation Force XII
D MATLAB Codes XV
D.1 Main script . . . XV D.2 Mass and Hydrostatic Calculations . . . XVII D.3 Morison’s Equation . . . XX D.4 Calculation of Internal Hull Loads . . . XXII
2.2 Existing semi-submersible FWT designs I . . . 5
2.3 Existing semi-submersible FWT designs II . . . 6
2.4 Drag coefficients for circular heave plates . . . 12
4.1 Illustrations of the WindFloat concept . . . 22
4.2 Screenshots of the WindFloat model in GeniE andHydroD . . . 24
4.3 Split force between pontoons . . . 25
4.4 Wave spectra for the three environmental conditions . . . 30
4.5 Definition of righting moment and wind heeling moment curves . . . 32
4.6 Mesh study of the 5 MW WindFloat model . . . 38
4.7 RAOs for surge, heave and pitch motion for the 5 MW WindFloat structure . . . 40
4.8 Verification of procedure for calculating internal loads . . . 41
5.1 Wave loads on the upscaled design . . . 44
5.2 Comparison of heave potential excitation force calculated byWadam against the estimated heave force . . . 44
5.3 Added mass and damping terms for the upscaled design from Wadam and Mori- son’s equation . . . 45
5.4 Sectional forces and moment amplitudes for the upscaled design . . . 47
5.5 Effect of column diameter on steel mass, pitch restoring coefficient, heave natural period, and pitch inertia. . . 49
5.6 Added mass and damping coefficients for different column diameters . . . 50
5.7 First-order wave excitation load amplitudes per unit wave amplitude in surge, heave, and pitch for different column diameters . . . 51
5.8 Surge wave excitation force calculated by Wadam and with Morison’s equation . 51 5.9 Surge, heave, and pitch RAOs for different column diameters . . . 53
5.10 Sectional force and moment amplitudes for different column diameters . . . 54
5.11 Effect of changing draft on steel mass, pitch restoring coefficient, heave natural period, and pitch inertia. . . 55
5.12 Added mass and damping coefficients for different drafts . . . 56
5.13 First-order wave excitation load amplitudes per unit wave amplitude in surge, heave, and pitch for different drafts . . . 57
5.14 Surge, heave, and pitch RAOs for different drafts . . . 58
5.15 Sectional force and moment amplitudes for different drafts . . . 59
5.16 Effect of changing DCC on steel mass, pitch restoring coefficient, heave natural period, and pitch inertia. . . 60
5.17 Added mass and damping coefficients for different DCC . . . 61
5.18 First-order wave excitation load amplitudes per unit wave amplitude in surge, heave, and pitch for different DCC . . . 62
5.19 Surge, heave, and pitch RAOs for different DCC . . . 63
5.20 Sectional force and moment amplitudes for different DCC . . . 64
5.21 Effect of changing LHP on steel mass, pitch restoring coefficient, heave natural period, and pitch inertia. . . 65
5.22 Added mass and damping coefficients for different LHP . . . 66
5.23 First-order wave excitation load amplitudes per unit wave amplitude in surge, heave, and pitch for different LHP . . . 67
5.24 Surge, heave, and pitch RAOs for different LHP . . . 68
5.25 Sectional force and moment amplitudes for different LHP . . . 69
5.26 Sensitivity analysis of how the parameters affect the design criteria . . . 70
parameters are varied for different environmental conditions . . . 72 5.29 Relative change in the standard deviations of the cross section x- and z-force,
and bending moment when the parameters are varied for different environmental conditions . . . 72 5.30 Relative changes in the standard deviations of the motions, internal loads, and
nacelle accelerations for different designs . . . 76 5.31 Screenshot of the model from HydroD with a negative 10 degrees tilt angle . . . . 77 5.32 Added mass and damping coefficients for different tilt angles . . . 77 5.33 Coupled added mass and potential damping coefficients between surge, heave,
and pitch for different mean tilt angles . . . 78 5.34 Wave excitation load amplitudes in surge, heave, and pitch for different tilt angles 79 5.35 A comparison of the surge wave excitation force amplitudes calculated byWadam
with 10 degrees tilt against the inertia force on the columns with different tilt angles using Morison’s equation . . . 80 5.36 Surge, heave, and pitch RAOs for different tilt angles . . . 81 5.37 Sectional force and moment amplitudes for different tilt angles . . . 82 5.38 Relative changes in the standard deviations of the motions, internal loads, and
nacelle accelerations for different tilt angles . . . 83 A.1 Added mass coefficients for the upscaled 10 MW platform . . . I A.2 Potential damping coefficients for the upscaled 10 MW platform . . . I A.3 Pitch RAO when solved with the one DOF and six DOF equations of motion . . II B.1 Normalized added mass and damping for different column diameters . . . II B.2 Normalized wave excitation loads in surge, heave, and pitch for different column
diameters . . . III B.3 Morison wave loads plotted for different column diameters . . . III B.4 Morison added mass and damping plotted for different column diameters . . . IV B.5 Normalized internal forces in x and z, and bending moment in cross section for
different column diameters . . . IV B.6 Internal forces and moments in cross section for column diameter equal 12.8 and
15 meters . . . IV B.7 Normalized added mass and damping in surge and yaw for different drafts . . . . V B.8 Normalized wave excitation loads in surge, heave, and pitch for different drafts . V B.9 Morison wave loads plotted for different drafts . . . VI B.10 Morison added mass and damping plotted for different drafts . . . VI B.11 Internal forces and moments in cross section for draft equal to 14 and 24 meters . VI B.12 Added mass and damping in pitch and yaw for different DCC . . . VII B.13 Morison wave loads plotted for different DCC . . . VII B.14 Morison added mass and damping plotted for different DCC . . . VII B.15 Internal forces and moment in cross section for DCC equal to 57 and 73 meters . VIII B.16 Normalized added mass and damping in heave and pitch for different LHP . . . . VIII B.17 Normalized wave excitation forces in surge, heave, and pitch for different LHP . . VIII B.18 Morison wave loads plotted for different LHP . . . IX B.19 Morison added mass and damping plotted for different LHP . . . IX B.20 Internal forces and moment in cross section for LHP equal to 18 and 24 meters . IX B.21 Morison added mass and damping plotted for different tilt angles . . . X B.22 Internal forces and moment in cross section for zero and ±10 degrees tilt angle . XI
2.2 Scale factors for different parameters . . . 9
4.1 Main parameters of the DTU 10 MW Reference Wind Turbine . . . 29
4.2 Tower properties given for the LIFES50+ 10 MW structures . . . 29
4.3 Environmental conditions defined . . . 30
4.4 Added mass and drag coefficients used for the analyses . . . 36
4.5 5 MW WindFloat main dimensions . . . 37
4.6 5 MW WindFloat assumed dimensions . . . 37
4.7 Comparison of hydrostatic data for 5 MW WindFloat . . . 39
4.8 Comparison of natural periods and added mass and inertia for 5 MW WindFloat 39 5.1 Upscaled 10 MW WindFloat main dimensions . . . 42
5.2 Main design criteria for the upscaled design . . . 43
5.3 Frequencies of interest for the internal load calculations . . . 48
5.4 Comparison of how the parameters influence the platform behavior . . . 73
5.5 Improved 10 MW WindFloat main dimensions . . . 74
5.6 Main properties of the improved design . . . 75
CFD computational fluid dynamics
COB center of buoyancy
COG center of gravity
DCC distance between center of columns
DOF degree of freedom
EC environmental condition
FWT floating wind turbine
HP heave plate
IEA International Energy Agency
IRENA International Renewable Energy Agency
KC Keulegan-Carpenter
LHP length of heave plate edge
MSL mean sea level
MW megawatt
RAO response amplitude operator
RNA rotor nacelle assembly
RWT reference wind turbine
TLP tension-leg platform
WT wind turbine
Greek Letters
η,η,˙ η¨ body displacement, velocity, acceleration matrix (6x1)
ηk,η˙k,η¨k body displacement, velocity, acceleration, k = 1, ...,6 (surge, sway, heave, roll, pitch, and yaw, respectively)
λ wavelength [m]
∇ volume displacement [m3]
ω wave frequency [rad/s]
ρ density [kg/m3]
σ standard deviation
θ phase angle [rad]
ζ incident wave profile [m]
Latin Letters
A added mass matrix (6x6)
B linear damping matrix (6x6)
C hydrostatic restoring matrix (6x6)
F force matrix (6x1)
A area [m2]
Ca added mass coefficient [-]
CD drag coefficient [-]
D diameter [m]
k wave number [rad/m]
M mass [kg]
rii radius of gyration [m]
Sw(ω) wave spectrum [m2s/rad]
Sx(ω) response spectrum [m2s/rad]
t time [sec]
zhub hub height above MSL [m]
Subscripts
a amplitude
c restoring
col column
cs cross-section
hp heave plate
I inertia
p pressure
T thrust
1 Introduction
1.1 Background
The world needs to move its energy supply from fossil to more sustainable energy sources.
Onshore wind energy has proven to be a competitive solution to today’s coal and gas industries.
However, onshore wind requires large areas and has a large footprint on the land. In recent years, the wind industry has been moving offshore, building bottom-fixed wind turbines in shallow waters, where space limitations are less and the wind conditions are more favorable.
In 2018, offshore wind had a total installed capacity of 23 GW, providing 0.3 % of global electricity supply. However, the International Energy Agency, IEA, published a report in October 2019 stating that offshore wind has the potential to generate more than 420 000 TWh per year worldwide, more than 18 times the global electricity demand today (IEA,2019). Of this, 80 % of the potential is from sites located in deep waters, above 60 meters, where floating wind turbines, FWTs, are said to be more economically feasible than bottom-fixed (Cruz and Atcheson, 2016).
Further, IEA (2019) claims that under current policies the offshore wind market will expand by 13 % per year and become a $1 trillion business by 2040.
In the search of higher exploitation of wind energy, the wind industry is moving further out in the oceans, and FWTs are being developed. With the huge potential of harvesting deep water offshore wind energy, there is an increasing interest in optimization of the hull form to minimize cost, while still satisfying important strength and safety requirements.
1.2 Scope and Objectives
Several promising concepts exist for FWT hulls. The semi-submersible design has proven to be a competitive concept, due to its low draft and simple, well-known mooring system. Therefore, the scope of this thesis is to investigate how changes in the hull design affect the overall wave- induced responses of a semi-submersible FWT design. For a parameterized hull shape designed for a 10 MW wind turbine, the effects of changes to the hull shape on the first-order response amplitude operators, RAOs, will be assessed. In addition to global motions, first-order wave loads within the hull will also be considered. This will give a better understanding of how different parameters influence the responses and how the hull could be optimized.
The objectives of this thesis are:
• Give an overview of existing semi-submersible floating wind turbine designs.
• Choose a favorable semi-submersible design and upscale it to support a 10 MW wind turbine.
• Calculate the wave-induced motion RAOs and internal hull loads on the parameterized platform.
• Discuss and compare the influence different design parameters have on the responses of the semi-submersible platform design.
• Suggest a cost-effective design while complying with strength and safety regulations.
• Assess the importance of including a wind-induced mean tilt angle in the analyses of the platform on the RAOs.
1.3 Structure of the Report
The thesis is structured as follows: First, the topic is introduced and the scope and objectives of the thesis are defined inSection 1. Secondly,Section 2provides an overview of the offshore wind industry, different semi-submersible designs, and related research on the topic. InSection 3, the theory used for the calculations is presented and Section 4 explains the methods and software used for obtaining the results of the thesis. The results are presented and discussed inSection 5.
First, the 10 MW upscaled design and general findings are presented inSection 5.1. The results from the parametric study are given in Section 5.2, and an improved design is suggested in Section 5.3. The effects of including a mean tilt angle are assessed and discussed in Section 5.4.
The conclusion and recommendations to further work are given inSection 6. At the end of the report is the Appendix.
2 Literature Review
The literature review presents a further introduction to the offshore wind industry. Existing semi-submersible FWT platform designs are presented and compared in Section 2.2. Research related to this thesis is summarized in Section 2.3. InSection 2.4and Section 2.5are upscaling procedures of FWT hulls and the effects of heave plates discussed, respectively.
2.1 Offshore Wind Industry
The offshore wind industry has the advantage of the open sea, giving stronger and more reliable winds than those onshore. This results in higher capacity factors for offshore wind turbines compared to onshore wind turbines. The capacity factor is the average output over the year relative to the maximum rated power capacity. In 2018, the average global capacity factor for offshore wind turbines was 33 %, while for onshore wind turbines it was 25 % (IEA, 2019). In addition, new offshore wind projects are assumed to have capacity factors even above 50 %.
The offshore wind marked grew by almost 30 % per year between 2010 and 2018 and the IEA predicts offshore wind to be competitive with fossil fuels within the next decade.
The first offshore wind farm was installed in Denmark in 1991, consisting of 11 turbines with a total capacity of 4.95 MW (Cruz and Atcheson, 2016). Since then, there has been a huge technological development, for instance with increasing turbine sizes. The average offshore wind turbine, WT, size grew from 1.6 MW in 2000 to 5.5 MW in 2018, and the industry is aiming for turbines with 15-20 MW capacity for 2030 (IRENA,2019). Larger turbines leads to overall lower costs of the electricity produced and higher capacity factors, which are advantages for offshore WTs, and especially FWTs as they have the potential to be scaled larger than onshore and bottom-fixed WTs.
2.1.1 Floating Wind Turbines
Floating wind turbines, FWT, are more economically feasible than bottom-fixed for depths larger than 50-60 meters (Cruz and Atcheson, 2016). Therefore, it is normal to divide the available waters into shallow water, depths below 60 meters, and deep water, depths from 60 to 2 000 meters. This thesis focus on FWTs, as the potential for harvesting wind energy in deep water is nearly limitless.
Figure 2.1: The most common floater types for wind turbines. From the left, semi-submersible, spar, tension-leg and barge platform (DNV GL AS,2018).
A FWT structure consists of a rotor nacelle assembly (RNA), a tower, a hull, and a mooring system. The FWT structures could be divided into four concepts based on how they achieve
necessary stability, see Figure 2.1. The designs are taken from the offshore oil and gas industry (Cruz and Atcheson,2016).
• Semi-submersible platform, buoyancy-stabilized by having a large water-plane area.
• Spar platform, ballast-stabilized by having a large draft.
• Tension-leg platform, TLP, mooring-stabilized by use of taut moorings.
• Barge platform, also buoyancy stabilized.
There also exist intermediate designs. The different concepts have their advantages and dis- advantages. The spar platform requires large depths for installation, towing, and operation, but the fabrication is simpler than for the TLP or semi-submersible designs. The spar and TLP have more suited natural periods than the semi-submersible, but the semi-submersible has simpler mooring than the TLP. The semi-submersible design is stable at low draft and can be fully assembled onshore (Cruz and Atcheson, 2016). Due to these factors, it is not necessarily one design that is better than the others, site conditions and turbine size could be determining factors for selecting a design.
FWTs is a newly developing industry. In 2017, Equinor installed the first operational floating wind farm, the Hywind Scotland, consisting of 5 spar FWTs with a total installed capacity of 30 MW (Equinor, 2020). And in October 2019, Equinor (2019) announced their next FWT farm, the Hywind Tampen, consisting of 11 spar FWTs with a total capacity of 88 MW. Due to gained experience and scale effects, Equinor aims to halve capital expenditure per MW for Hywind Tampen.
Several other designs have been proposed in recent years and several demonstration projects are underway. For instance, the company Ideol has developed a barge floating structure for FWTs.
In 2018, they installed the Floatgen in France with a 2 MW turbine and the Hibiki in Japan with a 3 MW turbine and are planning a 24 MW wind farm of 4 units outside France (Ideol, 2018).
Principle Power is another company in the FWT industry with its semi-submersible concept WindFloat that is further explained inSection 2.2.3 and is used as the baseline design for this thesis. They deployed a 2 MW prototype in 2011 outside the coast of Portugal (Principle Power, 2019). In late 2019, they installed the world’s largest FWT, with an 8.4 MW wind turbine. This was the first of three FWTs that will make up the WindFloat Atlantic wind farm outside Portugal with a total installed capacity of 25 MW (Principle Power, 2020). The WindFloat concept is also to be used for a 10 MW WT outside the south coast of France (Veselina Petrova,2019).
Even though the FWT structures are inspired by the oil and gas industry, there are some important differences in the design criteria, which will help reduce costs when designing FWTs.
These differences are (Cruz and Atcheson, 2016, pp. 7–8),
• FWTs are un-manned, giving lower risk to human life and several safety mechanisms are not required.
• The potential damages due to failure of a FWT is much lower than for an oil and gas production facility.
2.2 Existing Semi-Submersible Designs
The semi-submersible designs for FWTs are characterized by three to five primary columns, with the turbine either at the center or over one of the offset columns. They are fabricated from either steel or concrete and often designed with heave plates to reduce motions as explained in
Section 2.5. They have a relatively low draft and use standard catenary mooring lines. Due to the large water-plane area, wave loads will be relatively high, hence, bracing is often necessary to reduce the load effects (Cruz and Atcheson,2016). The following sections give a summary of five distinct, promising semi-submersible designs that exist today. A comparison of the different designs is presented inSection 2.2.6.
(a) OO-Star (Dr.techn.Olav Olsen, 2020)
(b) NAUTILUS (Yu et al.,2018) (c) WindFloat (Roddier, Peiffer, et al.,2011).
Figure 2.2: Three different existing semi-submersible FWT designs.
2.2.1 Olav Olsen OO-Star
The Olav Olsen OO-Star semi-submersible platform is developed by the company Dr.techn.
Olav Olsen as part of the research project LIFES50+. LIFES50+ was part of the European Union funded program Horizon 2020 (LIFES50+, 2020). One of the objectives of the project was to optimize and qualify two innovative substructure designs for 10 MW turbines.
The design is shown inFigure 2.2a. It consists of a star-shaped base pontoon, which connects a central column with three outer columns, separated by a 120 degrees angle. The wind turbine, WT, is mounted on the center column, letting the buoyancy force of the center column absorb most of the WT weight. A disadvantage with the design is it could be difficult accessing the WT since one has to maneuver in between the outer columns. The outer columns have heave plates at the bottom. The platform is designed with post-tensioned concrete and a catenary mooring system with three mooring lines will be used. Concrete is chosen as the material to increase the lifetime of the structure since concrete has better fatigue properties. On the other side, using concrete yields a higher volume displacement and total substructure mass. The side columns have a diameter of 13.4 meters and the distance from the center column to the outer column is 37 meters. Yu et al. (2018) give a detailed description of platform geometry and structural and hydrodynamic properties.
2.2.2 NAUTILUS
The NAUTILUS semi-submersible substructure is the other design from the LIFES50+project.
The design is shown inFigure 2.2b. It is a four-columns structure in steel with the WT placed in the center, making the platform symmetrical. The WT is not supported by a central column, giving large stresses in the cross-shaped structure connecting the outer columns. A quadratic ring pontoon is mounted between the columns at the bottom, functioning as a heave plate.
Notable about the design is that no bracing is used, simplifying construction. The WT is easily accessed from a ladder at one of the columns. Active seawater ballast is used to reduce mean trim from wind loads and a standard catenary mooring system with four mooring lines is utilized.
The columns have a diameter of 10.5 meters and the distance between them is 54.8 meters. Yu et al. (2018) and Galván et al. (2018) give detailed descriptions of the platform geometry and structural and hydrodynamic properties.
2.2.3 WindFloat
The WindFloat semi-submersible platform for FWTs is developed by Principle Power, Inc. The design is shown inFigure 2.2c. It is a three-columns steel structure with heave plates and the WT mounted on one of the columns (Principle Power, 2019). With the WT above one of the side columns, the structure is not symmetrical, but the weight of the WT is absorbed by the buoyancy force of the column. Further, installing the WT on top of the substructure becomes easier, since the lift is shorter. The WT is easily accessed from a ladder at one of the other columns. A summary of a feasibility study for the WindFloat concept is presented in the papers:
Roddier, Cermelli, et al. (2009), Cermelli et al. (2009) and Aubault et al. (2009). The first paper describes the design basis, the second a hydrodynamic analysis, and the third a structural analysis. Further, Roddier, Peiffer, et al. (2011) present the dimensions and hydrodynamic performance of the WindFloat concept equipped with a generic 5 MW wind turbine. The columns have a diameter of 10 meters and the distance between them is 46.0 meters.
The design utilizes four standard, catenary mooring lines. The columns contain permanent water ballast, to achieve the operational draft. An active internal water ballast system is used to move water between the columns to counteract the mean drag loads on the WT. The system has a 20 minutes reaction time, to only account for significant changes in wind speed and direction.
(a) Fukushima (Fukushima Off- shore Wind Consortium,2020)
(b) OC4-DeepCwind (Robertson et al.,2014)
(c) OC4-DeepCwind (Robertson et al.,2014).
Figure 2.3: Two different existing semi-submersible FWT designs. In (c), a picture of a 1/50 scale model of the OC4-DeepCwind support structure is given.
2.2.4 Fukushima
The Fukushima offshore wind consortium in Japan has designed a V-shaped semi-submersible support structure for a 7 MW offshore wind turbine. The design is shown inFigure 2.3a. It is a simple design with three columns mounted in a V-shape, whereas the turbine is mounted on top of the center column. The development of the design is described by Ohta et al. (2013). As for the WindFloat, mounting the WT onto the platform requires a shorter lift than for the other designs. The platform consists of only flat surfaces, making manufacturing simple and mass production cheap. The design is also without bracing, lowering the risk of fatigue damages, and simplifies construction. Seawater ballast is used for regulating the draft and the trim of the platform due to mean drag forces on the WT. The platform is moored with 8 pieces of catenary
mooring. Karimirad and Michailides (2015) also report that using a braceless V-shaped semi- submersible platform for FWT is a feasible solution. Internal forces could become a problem as the two end columns are only mounted to the center column, giving possibly large bending moments on the pontoons.
2.2.5 OC4-DeepCwind
The OC4-DeepCwind semi-submersible is designed by the DeepCwind consortium for the Off- shore Code Comparison Collaboration Continuation, OC4, project. The design is shown in Figure 2.3b and Figure 2.3c. It is designed for a 5 MW wind turbine, which is mounted on a center column. There are three offset columns that are connected with the center column through cross members. There are in total 15 members, six connecting the offset columns in a triangle form, and nine connecting them to the center column. This makes the structure complex and increases production costs. A base column is designed at the bottom of each offset column, to reduce motions of the platform in waves by working as heave plates. The platform is moored with three catenary mooring lines and passive water ballast is used to obtain the desired draft.
The structural and hydrodynamic properties of the platform are described by Robertson et al.
(2014).
2.2.6 Comparison of Existing Designs
In Table 2.1 are the five FWT semi-submersible designs compared with respect to different parameters.
Table 2.1: Comparison of the five semi-submersible FWT designs. The substructure mass is without tower and mooring mass, but with ballast. Displacement is given for operation.
Parameter OO-Star NAUTILUS WindFloat Fukushima OC4
WT Power [MW] 10 10 5 7 5
Number of columns [-] 4 4 3 3 4
Material [-] Concrete Steel Steel Steel Steel
Hull material mass [kg] - 2.70E+06 - - 3.85E+06
Substructure mass [kg] 2.17E+07 6.58E+06 3.94E+06 - 1.35E+07
Displacement [m3] 23 509 8 113 4 527 26 000 13 917
Draft [m] 22 15 17 17 20
2.3 Related Studies
Karimi et al. (2017) carried out a multi-objective design optimization for offshore FWT sup- port structures to support a 5 MW WT. The optimization process builds on the work by Hall et al. (2013), but with a new optimization algorithm and an updated frequency domain dynamic model, which changed the Pareto optimal platform designs. First-order wave loads on three types of parameterized platform structures, tension-leg, spar buoy, and semi-submersible, were calculated usingWAMIT. Viscous effects were accounted for by using a linear representation of Morison’s equation. Changing the design parameters for the three types of structures, they calcu- lated and compared the cost of construction and mooring system against the standard deviation of the nacelle fore-aft accelerations. The study showed that a four-column semi-submersible and tension-leg structures are more advantageous than a spar design. For the cheapest possible plat- form, the four-column semi-submersible platforms gave the best combination with low nacelle accelerations. The TLP-platforms could give lower nacelle accelerations, but then at a higher cost.
The semi-submersible design space consisted of a center column and three to five outer columns
with heave plates. The results showed a negative relationship between nacelle accelerations and overall cost, as for the other platforms. The four-column designs gave the most optimal combinations of low nacelle accelerations and low cost. For the semi-submersible optimal design points, reduced nacelle acceleration, at the price of higher cost, was achieved by the trend of increasing the draft, reducing the column diameters, and increasing the heave plate diameter.
The draft and the heave plate diameter were increased by a higher factor than the reduction in column diameter. The radius from the center column to the outer columns was constantly around 28 meters for all the optimal design points. For all the four-column optimal design points an angled taut mooring system was suggested. While for the five and six columns designs, slack catenary mooring was used but moved towards taut mooring for more expensive designs. For the five and six columns designs, for achieving lower nacelle accelerations the heave plates were increased more in size, while the draft was adjusted less, implying that the heave plates were most important in obtaining lower nacelle accelerations. The outer column diameters were also increased, but much less than the heave plates.
Hall et al. (2014) performed an optimization process for FWT platforms by combining and linearly interpolating the hydrodynamic coefficients of six basis platforms, to explore new pos- sible designs. The calculations were carried out in the frequency domain, with the objective of minimizing the nacelle accelerations. The results for a slack catenary mooring system and tension-leg mooring system gave similar results, with lowest nacelle accelerations by combining a large submerged volume with a widely distributed waterplane area.
Tracy (2007) presents a parametric study of the design space for a FWT structure. The pa- rameterized structure consisted of a concrete ballasted cylinder, with three types of mooring configurations, slack catenary, tensions-leg, and taut catenary. Also here, linear hydrodynamics andWAMIT were used for calculating the hydrodynamic coefficients and the motions were cal- culated in the frequency domain. The standard deviations of the nacelle accelerations were com- pared against mooring line tensions and platform displacement. For all of the above-mentioned analyses, Pareto optimal designs are presented, where several parameters are adjusted for each design. Therefore, the influence each parameter has on the design is not studied, only the overall trends.
Wang (2014) presents the design process of a pontoon-type semi-submersible platform designed to support a 10 MW wind turbine. The design is based on the 5-MW-CSC hull presented by Luan, Gao, et al. (2016). The motions of the platform and internal loads were calculated using first-order potential theory and Wadam. Second-order wave loads were calculated, a mooring system proposed, and coupled wind-wave dynamic analysis performed. Also, a sensitivity anal- ysis of how the different main dimensions affect the structure mass, displacement, maximum static pitch angle, and heave natural period was carried out based on hand calculations and used for the initial design. However, the effects on the hydrodynamic coefficients and excitation loads were not discussed, as in this thesis. The draft of the 10 MW design was decreased to 20 meters to make a less over-conservative design. The column diameters and distance between them were increased to account for the increased overturning moment from the WT.
The doctoral thesis by Bachynski (2014) deals with the design and dynamic analysis of FWT tension-leg platforms. Among other things, the effects of changing different design parameters on the behavior of FWT tension-leg platforms were investigated. The results from this investigation are also presented in Bachynski and Moan (2012). The designs were analyzed in waves and wind, using first- and higher-order theories. Even though only first-order waves are included in this thesis, the work by Bachynski (2014) is used as inspiration for how to present the results of the parametric study.
Antonutti et al. (2014) studied the effect of a wind-induced mean tilt angle on the hydrodynamics of the Dutch Tri-floater semi-submersible FWT design with heave plates. The influence of the submergence of the heave plates was also investigated. The analyses were conducted using first-order potential theory, and a boundary element solver. The study showed that the mean tilt angle could have important effects on the hydrodynamic coefficients and excitation forces, especially due to the vertical displacement of the heave plates and that this should be accounted for. The study was extended to analyses in the time domain in Antonutti et al. (2016), also showing that the mean tilt angle can affect the dynamics of the structure.
2.4 Upscaling Procedure
When upscaling a structure, geometric self-similarity is a usual procedure to use. A scaling factor,k, is then defined as the ratio of lengths,
k= lupscaled linitial
, (2.1)
which is used to upscale all dimensions. Following this procedure, the scale factors for other relevant parameters are given inTable 2.2.
Table 2.2: Scale factors for different relevant parameters using geometric self-similarity scaling laws.
CP is the power coefficient of a wind turbine andvwind is the velocity of the wind.
Parameter Equation Scale factor
Length l k
Volume V =l3 k3
Mass M =ρV k3
Power P = ρair2 CpπDrotor24 v3wind k2
When upscaling a wind turbine, a standard procedure is choosing a scaling factor based on the power rating of the upscaled wind turbine and the base wind turbine design, in other words, k = p
PW T,upscaled/PW T,initial (Bak et al., 2013). However, Leimeister (2016) presents an upscaling procedure for semi-submersible floaters when designed for a larger wind turbine.
Instead of the scaling factor being based on the power ratio, Leimeister suggests it should be based on the mass ratio as,
k= 3
sMW T,upscaled
MW T,initial
. (2.2)
Following scaling theory, seeTable 2.2, this should give the same scale factor as using the power ratio. However, due to technological developments in materials and efficiency, larger, new WT designs are improved with respect to the weight per power-output.
Leimeister (2016) based her work on the DeepCwind OC4 platform. The platform is an over- conservative suggested design, with a large mass and low maximum static pitch angle. It also has a low natural period in heave. Therefore, Leimeister first suggests an optimized design, where the upper side column diameters were scaled down, resulting in lower mass, higher maximum pitch angle, and higher heave natural period. The downscaling was based on obtaining a heave natural period of at least 20 seconds and a maximum static pitch angle of 5 degrees. The optimized design showed a reduction in the heave standard deviation.
For an optimized upscaling of a semi-submersible structure, Leimeister suggests using a different scaling factor on the diameters and wall thicknesses of the columns breaking the free surface.
The center column diameter should scale with the diameter of the WT tower, so they still have the same diameter at the connection point. For the side columns, a scale factor that maintain the static pitch angle of the initial design is proposed, since it is assumed to be critically optimized for the initial design.
A design for the WindFloat concept supporting a 10 MW WT is presented by Son et al. (2018).
The diameter of the columns is kept at 10 meters, the distance between the columns is increased to 70 meters, and the lightship weight is 2000 tons. More information is not given, as this is the property of Principle Power. The work presents RAOs and nacelle accelerations without axes for comparing results from different software.
2.5 Effects of Heave Plates
Heave plates are popular appendages used on semi-submersible and spar platforms. Heave plates provide the structure with increased added mass and damping in heave, roll, and pitch. The increase in added mass comes from the large amount of water that is displaced as the platform moves. Therefore, heave plates are often called water entrapment plates. The increase in added mass gives a higher natural period in heave, roll, and pitch, moving the natural periods out of the wave energy spectrum. Also, the sharp edges of the heave plates increase vortex shedding which gives a larger damping force and reduces the platform motions (Roddier, Peiffer, et al., 2011).
Due to the highly non-linear problem of heave plates, since viscous forces are important, experi- ments and CFD calculations are necessary for estimating the added mass and damping. Several studies have been conducted about the subject, as heave plates have been commonly used in the oil and gas industry, and are popular in the design of FWT support structures. When comparing results from calculations on heave plates, two non-dimensional parameters are often used, the Keulegan-Carpenter (KC) number andβ,
KC= 2πη3 Dhp
& β = D2hpf
ν , (2.3)
whereη3 is the heave amplitude of oscillation, Dhp is the diameter of the heave plate,f is the frequency of oscillation andνis the kinematic viscosity of the fluid. The Reynolds number could be obtained by Re=KC β.
The hydrodynamic properties of a heave plate depend on, among others, the plate thickness- diameter ratio, the ratio between the diameters of the heave plate and the column it is attached to, the porosity, and the oscillation amplitude and frequency. Numerical studies on a single column with a circular heave plate attached to the keel are presented by, for instance, Tao and K. Thiagarajan (2003), Tao and Cai (2004), and Tao, Molin, et al. (2007). The thickness- diameter ratio of the plate affects the generation of vortices, in the sense that for a very thin heave plate, the two edges of the plate act as one sharp edge, giving another flow field than for a thicker plate. The studies showed a trend of increasing added mass and damping for lower thickness-diameter ratios due to viscous effects and mainly for low ratios. The impact was largest on the viscous damping. The column attached to the heave plate will reduce the effect of the heave plate on the column side since less water will be entrapped, reducing mainly the added mass. The theoretical added mass of a circular disk oscillating vertically in infinite water is (DNV GL AS,2019a),
A33= 1
3ρD3hp. (2.4)
Tao, Molin, et al. (2007) suggest the theoretical added mass of a cylinder with a circular disk attached at its base could be estimated as,
A33= 1
12ρ(2Dhp3 + 3πDhp2 z−π3z3−3πDcol2 z), (2.5) where z= π1q
D2hp−D2col, and Dcol is the diameter of the column. Tao and Cai (2004) found that the damping also increases with increasingDhp/Dcol, but tends to flatten for higher ratios and that this value depends on the KC number. For KC equal 1, 0.5, and 0.1, the flattening occurs at Dhp/Dcol ≈1.65, Dhp/Dcol ≈1.45, and Dhp/Dcol ≈1.3, respectively. At low ratios, the boundary layer from the columns reduce the vortex shedding at the edges, lowering the damping.
Tao and Dray (2008) found through model tests that heave plates with high porosity increase the damping at very low KC numbers, while it, in general, gives lower added mass for all KC numbers. When the oscillation amplitude goes to zero, the added mass from the heave plates will also become zero, implying that added mass found from experiments is very dependent on the KC number. The added mass and damping coefficients were found to increase linearly with increasing KC numbers for the tested area of 0.2< KC <1.2. The quadratic drag coefficients for the heave plates when oscillated at 1 Hz are plotted for different KC numbers inFigure 2.4a.
The drag coefficient decreases non-linearly with increasing KC number.
All the above-mentioned experiments were conducted in deep water conditions, with the heave plate being deeply submerged. Wadhwa and K. P. Thiagarajan (2009) studied the effects of a disk oscillating close to the free surface experimentally, with submergence-diameter ratios from 0.1 to 1. Both added mass and damping coefficients increased with lower submergence, and a good agreement with the results from Tao and Dray (2008) was shown for the submergence-diameter ratio of 1, meaning this could be assumed as the heave plate being deeply submerged.
Lopez-Pavon and Souto-Iglesias (2015) studied the effects of heave plates on a three-column semi- submersible FWT. Model tests of one of the columns with a plain and a reinforced heave plate were completed. The results were compared to numerical simulations. Both a frequency domain first-order panel method in Wadam and a Reynolds-Averaged Navier-Stokes CFD method in ANSYS CFX were used. The results for the added mass and damping coefficients showed a weak link with the oscillation frequency and a large dependence with the KC number. The low dependency of frequency is explained with the high submergence-diameter ratio of the heave plate, being 0.775. Further, for a plain disk the Wadam calculated added mass was close to the theoretical value for a column-plate structure, equation (2.5), also with a low dependency of frequency. The added mass from experiments and CFD showed good agreement and were higher than the theoretical value and increased with higher KC numbers. A higher KC number gave a higher heave damping coefficient also, as found by Tao, Molin, et al. (2007) and Tao and Dray (2008). The potential radiation damping fromWadam was found to be negligible, again because of the relatively large distance to the surface. The reinforced plate gave higher added mass than the plain for low KC numbers and in Wadam, while the damping of the reinforced plate was lower for all KC numbers since the vortex shedding was reduced. The differences in added mass were much lower than for the damping. The drag coefficients found from the experiments are plotted for different frequencies and KC numbers inFigure 2.4b.
The mentioned studies have been conducted with circular heave plates. Moreno et al. (2016) present experimental results for the heave added mass and damping of a column-heave plate structure with a circular and a hexagonal heave plate made for FWT platforms. The hexagonal
heave plate was based on the WindFloat concept. The comparison showed that a circular and hexagonal heave plate of the same equivalent diameter have similar hydrodynamic properties over the range of 0.05 < KC <1.2 and 1 Hz < f < 6 Hz tested. The largest difference found was 8 % in damping. Further, the results showed that the added mass and damping coefficients are very dependent on β for KC < 0.3, where increasing frequency gave higher added mass and damping. The submergence-diameter ratio of the heave plates was 0.9. This trend could also be seen in Figure 2.4b for KC = 0.309, based on the experiments by Lopez-Pavon and Souto-Iglesias (2015).
Figure 2.4 shows the quadratic drag coefficients found with model experiments by Tao and Dray (2008) and Lopez-Pavon and Souto-Iglesias (2015) for circular heave plates. As seen from the figure, the drag coefficients from the two different experiments are similar for the same KC numbers. Even though the experiments were completed with models of different Dhp/Dcol ratio, thickness-diameter ratio, submergence-diameter ratio, and frequency, implying that the KC number is the most important parameter in the tested range.
(a)Tao and Dray (2008) (b) Lopez-Pavon and Souto-Iglesias (2015) Figure 2.4: Drag coefficients for circular heave plates from experiments. The results from Tao and Dray (2008) are given for a oscillation frequency of 1 Hz and for heave plates of different porosity. The results from Lopez-Pavon and Souto-Iglesias (2015) are given for a frequency made non-dimensional with the theoretical heave natural frequency. The drag coefficients are given for different KC numbers, and a plain and reinforced disk.
To summarize, the potential added mass is highly influenced by theDhp/Dcol ratio. As found by Lopez-Pavon and Souto-Iglesias (2015), modeling heave plates in potential theory will give errors, as viscous effects change the pressure field around the heave plates. The potential added mass calculated for the heave plates was lower than found from experiments. Using experimental data and potential calculations one could calculate a correction of added mass for the heave plates.
One could also find parametric data for heave plates, and use this for correcting the added mass.
However, this demands to find data for a heave plate with the same geometrical parameters and in the same conditions, as discussed above. Since a parametric study is conducted in this thesis, the potential added mass from Wadam will be used without correction to simplify the procedure.
Potential damping could be neglected, at least if the submergence-diameter ratio is above 1. For these cases, viscous drag loads from the vortex shedding at the edges are most important. The drag coefficient is mainly dependent on the thickness-diameter ratio of the heave plate and the KC number. For low KC numbers, the Dhp/Dcol ratio and the oscillation frequency could also be important. As seen from the comparison inFigure 2.4, the KC number seems to be the most important factor in the tested range.
3 Theory
This section will describe the theory used for the calculations. First, the rigid body equations of motions and how they are solved in the frequency domain are described. Secondly, the hydrodynamic theory used is presented. The coordinate system follows the right-hand rule, withz= 0 at the mean sea level and the z-axis pointing upwards.
3.1 Structural Dynamics
3.1.1 Rigid Body Equations of Motions
In this thesis, it is assumed that the platform with the wind turbine behaves as a rigid body.
Further, by assuming steady-state harmonic responses, the coupled equations of rigid body motions for the floating platform in six degrees of freedom are given as (Faltinsen, 1990, p. 66),
6
X
k=1
[(Mjk+Ajk(ω))¨ηk+Bjk(ω) ˙ηk+Cjkηk] =Fj(ω)eiwt; for j = 1, ...,6. (3.1) Whereηk are the motions, theksubscript denotes the mode of the six degrees of freedom, being surge, sway, heave, roll, pitch and yaw in the increasing order ofk, respectively, and the dot(s) represent time derivative(s). The j subscript denotes the force direction. The equations for j= 1,2,3 results from Newton’s second law, while the equations forj= 4,5,6 follows from the equation of angular momentum. Mjk are the components of the mass and inertia matrix for the structure,Ajk are the added mass and inertia coefficients, Bjk are the damping coefficients,Cjk are the hydrostatic restoring coefficients,Fj are the complex amplitudes of the excitation forces and moments on the structure and ω is the frequency of the harmonic excitation. The mass matrix is defined in this section, while the other terms are further explained inSection 3.2.3and Section 3.2.4.
The uncoupled and undamped natural frequency,ωnk of thekdegree of freedom for the platform is given by,
ωnk =
r Ckk
Mkk+Akk. (3.2)
Assuming coupling effects do not significantly modify the natural frequency, excitation forces at this frequency will result in large motions unless the damping is high or the excitation force is low due to for instance cancellation effects.
The mass matrix for a floating structure with the center of gravity located at (xG, yG, zG) is given as,
M =
M 0 0 0 M zG −M yG
0 M 0 −M zG 0 M xG
0 0 M M yG −M xG 0
0 −M zG M yG I44 I45 I46
M zG 0 −M xG I54 I55 I56
−M yG M xG 0 I64 I65 I66
. (3.3)
WhereM is the total mass of the structure, andIkk are the moments of inertia andIjk are the products of inertia with respect to the coordinate system(x, y, z). The moment of inertia terms are defined as,
